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May 18, 2005

String(n), Part I

Posted by Urs Schreiber

I was asked to say something about the meaning of the group String(n) and about manifolds with string structure.

So here I’ll try to give a somewhat more comprehensive discussion than last time that we talked about this.

First recall the situation for Spin(n).
A (Riemannian) manifold Mis spin or admits a spin structure if spinning particles can consistently propagate on it.

This is the case iff an SO(n)-bundle

(1)E↓M

over the manifold can be lifted to a Spin(n)-bundle, where Spin(n) is (of course) the central extension of SO(n) by ℤ2:

(2)1→ℤ2→Spin(n)→SO(n)→1.

And this is the case iff M is orientable and the second Stiefel-Whitney class w2(E)∈H1(M;ℤ/2) vanishes.

The situation for String(n) is similar, but with everything lifted by one dimension.
A manifold is string or admits a string structure if spinning strings can consistently propagate on it.

This is the case iff a principal loop-group LSO(n)-bundle

(3)LE↓LM

over the free loop space LM can be lifted to a LSO(n)^-bundle, where LSO(n)^ is a (Kac-Moody-)central extension of LSO(n) by U(1):

(4)1→U(1)→LSO(n)^→LSO(n)→1.

And this is the case iff the so-called string class of LM in H3(LM;ℤ) vanishes.

These two conditions on the topology of LM can equivalently be formulated in terms of M itself:

1) The vanishing of the string class in H3(LM;ℤ) is equivalent to the vanishing of the Pontryagin class 12p1(E) of any vector bundle associated to a principal Spin(n)-bundle E→M.

The string class in H3(LM;ℤ) is obtained from the Pontryagin class p1/2 by transgression. This means that it is represented by the 3-form

(5)∫γev*(ξ),

where ξ is a representative of p1/2, ev* is the pull-back by the evaluation map

(6)ev:LM×S1→M(γ,σ)↦γ(σ)

and ∫γ denotes the integral over the S1-factor in LM×S1.

2) This again is equivalent to the existence of a lift of the structure group of E from Spin(n) to the topological group called String(n).

The group String(n) (or rather a ‘realization’ thereof) is defined as a topological group all of whose homotopy groups equal those of Spin(n), except for the third one, which has to vanish for String(n):

(7)πk(String(n))={πk(Spin(n))fork≠31fork=3

Why this makes sense is best made plausible by looking at the first few homotopy groups of O(n). For n>8 they start as follows:

(8)k=01234567πk(O(n))=ℤ/2ℤ/20ℤ/2000ℤ

We get from O(n) to SO(n) by ‘killing’ the 0th homotopy group, i.e. by going to the connected component.

We get from SO(n) to Spin(n) by ‘killing’ the 1st homotopy group, i.e. by going to the universal cover.

We get from Spin(n) to String(n) by ‘killing’ the next nonvanishing homotopy group, which is the 3rd.

Since every simple Lie group has π3=ℤ it follows that Spin(n) cannot be a Lie group. It is only a topological group. (Meaning that it is a topological space on which the group multiplication acts as a continuous map, but that there is no smooth structure with respect to which the group operation were smooth.)

It can be shown and is well known that an equivalent way to define (a realization of) the group String(n) is as the topological group which makes this sequence of groups exact:

(9)1→K(ℤ,2)→String(n)→Spin(n)→1.

Here K(ℤ,2) denotes (a realization of) the Eilenberg-MacLane space K(ℤ,2), which is by definition a topological space all whose homotopy groups vanish, except for the second one, which is isomorphic to ℤ. In general

(10)πk(K(G,n))≃{Gfork=n1otherwise,

by definition.

The importance of string structures in string theory results from the fact that superstrings are nothing but ‘spinning strings’, i.e. fermions on loop space, and that their quantum equations of motion are nothing but a generalized Dirac equation on loop space. (The 0-mode of the worldsheet supercharge is a generalized Dirac(-Ramond) operator on loop space (for the closed string).)

It hence follows by the above discussion that superstrings can propagate consistently only on manifolds which are string, just like an ordinary point-like fermion can propagate consistently only on a manifold that is spin.

More technically, the wavefunction of a point-like fermion is really a section of a SO(n)^≃Spin(n)-bundle and hence such a bundle needs to exist over spacetime in order for the fermion to exists.

Similarly, the wavefunction of a fermionic string (‘spinning string’) is really a section of a LSO(n)^-bundle over loop space, and hence such a bundle needs to exist over the loop space over spacetime for fermionic strings to exist.

For instance the worldsheet supercharge of the heterotic string is a Dirac operator on loop space for fermions that are also ‘charged’ under an SO(32)- or E8×E8-bundle

(11)V↓M.

In K-theory one can form the difference bundle

(12)E=V−T,

where T is the tangent bundle and the condition for this bundle to admit a string structure is that the Pontryagin class vanishes, i.e. that

(13)p2(V)−p1(T)=0.

This is in fact the relation which follows from the cancellation of the perturbative anomaly of the effective SO(32)- or E8×E8-field theory obtained from these strings. Hence this famous anomaly is related to the fact that heterotic strings are spinors on loop space.

Lie Description of String(n)

If String(n) is not a Lie group, does it mean that it has no Lie algebra which can be used to describe it locally?

Yes.

I would find an explicit description in terms of Lie brackets illuminating.

String(n) itself cannot have such a description, directly. But the result which I reviewed in part II is something like the next best thing one can have.

This result says that the unwieldy, non-Lie group String(n) is what you get by starting with a nice Lie 2-group 𝒫1Spin(n) and taking its ‘nerve’.

The strict Lie 2-algebra of 𝒫1Spin(n) is easily written down. It consists just of the Lie algebra of paths in 𝔰𝔬(n), the Lie algebra of the Kac-Moody central extension of level 1 of loops ℓi in 𝔰𝔬(n):

(1)[(ℓ1,c1),(ℓ2,c2)]=([ℓ1,ℓ2],∫02π⟨ℓ1,ℓ′2⟩)

as well as the obvious action dα(p) of paths p on centrally extended loops (ℓ,c)

(2)dα(p)(ℓ,c)=([p,ℓ],∫02π⟨p,ℓ′⟩).

Here p:[0,2π]→𝔰𝔬(n)
is a based path, ℓ:[0,2π]→𝔰𝔬(n) a based loop, the brackets of these are pointwise brackets in 𝔰𝔬(n) and ci∈Lie(ℝ) are real numbers coming from the central extension. ⟨⋅,⋅⟩ is the Killing form of 𝔰𝔬(n) with a certain normalization chosen.

This infinite-dimensional Lie 2-algebra turns out to be equivalent to a very simple but non-strict Lie 2-algebra called 𝔰𝔬(k=1)(n), namely that consisting just of 𝔰𝔬(n) and the trivial Lie(ℝ) with trivial action of the first on the latter, but with a slightly non-trivial Jacobiator given by
J(x,y,z)=⟨x,[y,z]⟩ which relates the two sides of the Jacobi identity (which does hold) by an additional isomorphism.

So there is a path from a very manageable semistrict Lie 2-algebra

(3)𝔰𝔬(k=1)(n)

to the infinite dimensional but strict Lie 2-algebra

(4)𝒫1𝔰𝔬(n)

to the associated strict Lie 2-group

(5)𝒫1Spin(n)

to the unwieldy topological group

(6)String(n).

In a sense, all of the information about String(n) is encoded already in 𝔰𝔬(k=1)(n).

Read the post Some ConferencesWeblog: The n-Category CaféExcerpt: A conference on bundles and gerbes, another one on topology, and comments on associated 2-vector bundles and String connections.Tracked: April 19, 2007 8:57 PM

Read the post String- and Fivebrane-StructuresWeblog: The n-Category CaféExcerpt: A new article on String- and Fivebrane structures and some previous articles on Fivebrane structures.Tracked: October 14, 2008 10:15 PM